ON THE SQUARE BAR MICROMETER. 



171 



"1 f I 



and that to this must be added ^"^ g </, to get the time of crossing the hour 

 circle passing through the centre of the square. Consequently the true differ- 

 ence of right ascension of a comet and star will be 



= J ('.+ Q ~ i ft + k) +V^i W - <*) T f' * I] i (24) 



a — a 



whence the correction to be added to the uncorrected difference of right ascen- 



sion obtained by (1) is 



p sin 1 



15 cos 8 



// r 



2(<*W)-( ± £ T i)] 



(2.-,) 



or the equally convenient form 



f aiii i" [ * jjfffi' T y t - ro * ft- <,)]. (2 



To get the accurate expression for the difference of declination it is to he 

 noted that the distance from the angle of the square to the intersection of the 

 star's path with the diagonal is ±2 — dsecj>; also that the triangle formed by 

 the sides of the square and the star's path gives by Chauvenet's Trigonometry, 

 equation (282), 



±f — dsecp = ± (f 2 — ^) 15 cos S sin (45° + p) sin (45° —p); 



whence 7 , 9 * 2 — * i -, K s cos2/> 



d= ± r 2 cosp T — g- 15cos§-^> (2<) 



the difference of which and a similar equation for the comet gives the accurate 

 difference of declinations 



S'- 8 = (± £ T f) co S/ - 15 [± '*j* cosS' , ^cosSj^f • (28) 



This is a perfectly rigorous formula, whatever the value of p, and can be used 

 in any position of the square, should occasion arise, provided neither object 

 passes from the northern to the southern half of the square, or vice versa, dur- 

 ing its transit. In general, however, p is small, and we can develop cosy? and 



in series, neglecting 3 d and higher powers. Tims we obtain the correc- 

 tion to be added to the result found by (3), 



f rin» 1" @( * ^ * ^) 15 cos 8 -| (± f T 1)]. (20) 



or its equivalent. 



cos 2p . 

 cosp 



p 



» sin 2 1" [(± ^p=F ^) 15 cos 8 - J («T- </)]. (30) 



